In many research areas and industrial applications such as material analysis and science, quantum physics, nanophysics, semiconductor devices, there is an increasing demand for apparatuses and methods for effectively measuring small electrical signals in noisy environments.
One known technique for recovering such a signal of interest employs for instance lock-in amplification, which is a particular type of demodulation with specific requirements for the performance of the employed filter. The demodulation is performed on the measured wide-band signal to obtain the amplitude and the phase of the signal of interest at a specific frequency, i.e. narrow-band.
Another known method for recovering a small oscillation in a measured signal tracks the oscillation frequency e.g. by means of a phase-locked loop. Instead of demodulating always at the same frequency, phase-locked loops can track variations in the frequency of the measured signal, which further improves oscillation signal recovery.
Several applications require the measurement of a signal of interest at frequencies that are different from the fundamental frequency (also called center or carrier frequency) of the signal of interest, such as e.g. at the higher harmonics which are given by the fundamental frequency multiplied by an integer factor, or at the sideband frequencies, respectively. The sideband frequencies are defined as frequencies at a defined distance above or below the fundamental frequency. Sideband frequencies typically occur in applications where amplitude and/or frequency modulation techniques are employed. In the following the term “sidebands” is used for the expression “sideband frequencies”.
A known method for analyzing sidebands that occur in an amplitude or frequency modulated signal employs an apparatus 100 with two consecutive lock-in amplifiers, the apparatus 100 being depicted in FIG. 1. The apparatus 100 comprises a first mixer 102 for wide-band demodulation and a second mixer 104 for narrow-band demodulation of a measured input signal 101 with a fundamental frequency f2 and which has been modulated with a modulation frequency f1. The first mixer 102 has as incoming signals the measured input signal 101 and the output signal of a first oscillator 106 that has the frequency f2. In the first mixer 102 the input signal 101 is demodulated with the frequency f2 (wide-band demodulation). The output signal of the first mixer 101 is then low-pass filtered by a first low-pass filter 103 (LPF), the output signal of the first low-pass filter 103 having a spread spectrum around the frequency f2. The second mixer 104 has as incoming signals the output signal of the first low-pass filter 103 and the output signal of a second oscillator 107 with the frequency f1. In the second mixer 104 the output signal of the first low-pass filter 103 is demodulated at the frequency f1 (narrow-band demodulation). The output signal of the second mixer 104 is low-pass filtered by a second low-pass filter 105 (LPF). The output signal of the second low-pass filter 105 has the amplitude and the phase of the input signal 101 at the sidebands f2+f1 and f2−f1.
The known method employed by means of the apparatus 100 is known as spread spectrum demodulation or heterodyning in communication systems. It is also known as dual lock-in method in instrumentation. The term “spread spectrum demodulation” generally refers to any demodulation method that produces a signal with a spectrum that is much wider than the bandwidth of the signal of interest.
Another known example of an apparatus 200 employing spread spectrum demodulation is depicted in FIG. 2. The apparatus 200 comprises a mixer 203 with the input signal 201 and the output signal of an oscillator 202 as incoming signals. The mixer 203 produces as output signal a signal at a so-called intermediate frequency (IF). This signal is then low-pass filtered by a low-pass filter 206 (LPF) and further analyzed by using Fourier-transform-based methods by means of a Fourier-transform-unit 204 (FFT=fast Fourier transform). The first mixers 103, 203 depicted in FIGS. 1 and 2 often form part of a phase-locked loop for tracking the carrier/fundamental frequency.
An example of an application where the determination of sidebands is of interest is the analysis of the mechanical oscillation of a cantilever that is employed for surface potential measurements in scanning probe microscopy. For measuring surface potentials a cantilever is usually employed as mechanical oscillator to measure the surface topography as well as the electrostatic potential of a probe at atomic resolution. Typically a phase-looked loop is employed to track a resonance frequency f2 of the cantilever as it scans the surface of the probe. The cantilever is held close to its resonance frequency f2 by means of the phase-locked loop. Both amplitude and frequency modulation techniques are typically employed, i.e. the oscillation of the cantilever is amplitude or frequency modulated with a second frequency f1, with f1<f2. As a consequence of the modulation, sidebands occur at frequencies f2+f1, f2−f1 and possible other frequencies depending on the modulation method. The amplitudes of these sidebands are then measured to derive the surface potential. The standard method for measuring the amplitudes of the sidebands employs two consecutive lock-in amplifiers to perform a wide-band demodulation at the resonance frequency f2 and after that a narrow-band demodulation at the frequency f1 as shown in FIG. 1.
The disadvantage of the known methods is that they have two stages, i.e. two lock-in amplifiers or one lock-in amplifier and a Fourier-transform-unit. Hence, quite a lot of resources in form of signal processing units (mixers and filters) are necessary to analyze the input signal at a frequency of interest. Furthermore, with the known methods the amplitude and the phase can only be determined at sidebands in combination, i.e. it is not possible to look at each sideband separately.